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Chapter 8: Problem 74

Factor $$ -2 y^{2}-16 y t+40 t^{2} $$

Short Answer

Expert verified
The factored form is \(-2(y + 10t)(y - 2t)\).

Step by step solution

01

Identify a Common Factor

We begin by identifying any common factor in all terms of the polynomial \(-2y^2 - 16yt + 40t^2 \). Observing the coefficients (-2, -16, 40), we see that they are all divisible by \(-2\). Thus, we can factor out \(-2\) from the expression. This results in \(-2(y^2 + 8yt - 20t^2)\).
02

Factor the Quadratic Trinomial Inside Parentheses

Next, we focus on factoring the quadratic trinomial \(y^2 + 8yt - 20t^2 \) inside the parentheses. We look for two numbers whose product is equal to the constant term \(-20t^2 \) and whose sum is equal to the coefficient of the middle term \(8t\).Observing, we find that \(10t imes -2t = -20t^2\) and \(10t + (-2t) = 8t\). Thus, the trinomial factors to \((y + 10t)(y - 2t)\).
03

Write the Complete Factored Form

Substitute the factored trinomial back into the expression with the common factor we took out in the first step. This results in \(-2(y + 10t)(y - 2t)\). This is the completely factored form of the given polynomial.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Common Factor
Factoring polynomials often begins with identifying a common factor. A common factor is a number or variable that divides all terms in the expression without leaving a remainder. In the polynomial \[-2y^2 - 16yt + 40t^2,\] the numbers
  • -2,
  • -16,
  • 40
are coefficients of the terms. Each of these numbers can be divided by -2, which is the greatest common factor.
By extracting the common factor -2 from all terms, we simplify the expression without changing its value, resulting in: \[-2(y^2 + 8yt - 20t^2).\] This step reduces the complexity of the polynomial.
Factoring a Quadratic Trinomial
A quadratic trinomial is a polynomial of the form \[ax^2 + bx + c.\] To factor it, we look for two numbers that multiply to the constant term \(-20t^2\) and add up to the coefficient of the middle term \(8t\).

For the trinomial \[y^2 + 8yt - 20t^2,\] the goal is to break it into two binomials such that:
  • The product of the two numbers equals -20.
  • The sum of the two numbers equals 8.
After examining the possibilities, we find:
  • 10 times \(-2\) gives us \(-20\),
  • and \(10 + (-2)\) gives us 8.
Thus, the factors of the trinomials are: \[(y + 10t)(y - 2t).\] Factoring quadratic trinomials often requires identifying pairs of numbers that meet these conditions.
Forming the Factored Form
Once both the common factor and the quadratic trinomial have been identified, we can express the entire original polynomial in a factored form.

For the expression \[-2(y^2 + 8yt - 20t^2),\] substitute back the factored form of the quadratic trinomial: \[(y + 10t)(y - 2t).\]

Finally, incorporate the extracted common factor \(-2\) back into the expression to obtain the complete factored form: \[-2(y + 10t)(y - 2t).\]
This representation is more simplified and showcases how each polynomial term connects to its factors. Factored forms can make it easier to solve equations or to perform further polynomial operations.

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